Method for encoding a message into geometrically uniform space-time trellis codes

ABSTRACT

A method for coding messages by group codes over label alphabet and H-symmetric labeling is provided in order to encode geometrically uniform space-time trellis codes considering a distance spectrum. The coding method makes the complexity of code searching reduced considerably by limiting the object of the code design to geometrically uniform space-time trellis codes (STTCs). Therefore, it is possible to design the STTCs with many states, and also design the STTCs with superior performance.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a coding method that encodes messagesto be transmitted through a plurality of the transmit antennas in thecommunication system wherein signals are transmitted between a pluralityof transmit antennas and a plurality of receive antennas. Morespecifically, the present invention relates to a coding method forencoding a message with group codes over label alphabet and H-symmetriclabeling to encode geometrically uniform space-time trellis codesconsidering distance spectrum.

2. Description of the Related Art

Tarokh, et al., introduced space-time codes that were suitable formulti-input and multi-output antenna systems [V. Tarokh, N. Seshadri,and A. R. Calderbank, “Space-time codes for high data rates wirelesscommunication: Performance criterion and code construction”, IEEE Trans.Inform. Theory, vol. 47, pp. 744-765, March 1998]. Then, there has beenmuch research about designs and searches for good space-time codes. Mostof the space-time trellis codes (STTCS) are represented by modulatormapping functions and their codes over label alphabet, i.e., label code.

However, the label codes over label alphabet of the trellis codesaccording to prior arts are not group codes. As a result, space-timecodes in the prior art do not have geometrical uniformity, whichincreases the complexity of code searching.

Meanwhile, Firmanto, et al. designed a space-time trellis code with goodperformances for fast fading channels by using the design criteria whichminimize the maximum pair-wise error probability (PEP) [W. Firmanto, B.S. Vucetic, and J. Yuan, “Space-time TCM with improved performance onfast fading channels”, IEEE Commun. Lett., vol. 5, pp. 154-156, April2001]. However, as mentioned above, the design criteria which onlyminimize the maximum pair-wise error probability can not optimize theperformance of the space-time trellis code and can not predict theperformance precisely, either.

SUMMARY OF THE INVENTION

The inventors of the present invention have researched a message codingmethod which can decrease the complexity of code searching in amulti-input and multi-output communication system. As a result, theyhave found that geometrically uniform space-time trellis codes (GUSTTCs) for a fast fading channel can be constructed by group code overlabel alphabet and H-symmetric labeling, and that by searching for GUSTTCs, the complexity of code search can be extensively reduced.

Thus, it is possible to design the STTCs with many states by easilysearching a generator matrix G, which has been impossible previously.Further, according to the criteria considering the distance spectrum,the optimal STTCs with superior performance may be designed as well.

Therefore, the object of the present invention is to provide a codingmethod for coding a message to be transmitted through a plurality of thetransmit antennas into geometrically uniform space-time trellis codes(STTCs) in a multi-input multi-output (MIMO) communication system.

The present invention relates to a method for encoding a message whichis to be transmitted through multiple transmit antennas in thecommunication system wherein signals are transmitted between a pluralityof transmit antennas and a plurality of receive antennas. Morespecifically, the present invention relates to a method for encoding amessage with group codes over label alphabet and H-symmetric labeling.

The coding method of the present invention has been analyzed by modelingwhere there are fast fading channels between the MIMO antennas.

According to the present invention, messages are encoded into space-timetrellis codes (STTCs) by group codes over label alphabet and H-symmetriclabeling, and the encoded STTCs become geometrically uniform.

In order to code messages into geometrically uniform space-time trelliscodes according to the present invention, it is preferable to generatelabel code by matrix multiplying binary sequences of the messages bygenerator matrix G, at first. Then, the generated label codes are mappedover H-symmetric labeling to code the messages into codeword c which isa space-time trellis code.

More specifically, the binary sequence ū_(t) of the message at time t iscoded by the space-time trellis code c _(t) according to equation 1. ,c _(t)=M(ū _(t)

_(Ā) G ^(T))  [Equation 1]

-   -   wherein ū_(t) denotes the binary sequence in the alphabet A to        be modulated and transmitted through the transmit antenna,    -   G denotes a generator matrix,    -   the operation        _(Ā) denotes a matrix multiplication defined by multiplying and        adding operation of Ā,    -   Ā denotes the field whose algebraic structure under addition is        isomorphic to label alphabet A, and    -   M denotes an H-symmetric labeling mapping function.

In the present invention, it is preferable that the generator matrix isselected to maximize the minimum symbol hamming distance δ_(min) betweencodeword of random messages and all-zero codeword, and also maximize theeffective product distance at a predetermined number (such as δ_(min) orδ_(min)+1) symbol hamming distances.

More specifically, the generator matrix G is selected to minimize thevalue of equation 2.

$\begin{matrix}{{P_{f}(e)}_{N} = {\sum\limits_{\delta = \delta_{\min}}^{\delta_{\min} + N - 1}\;\left( {\left( {\sum\limits_{\Delta \in D_{\delta}}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}} \cdot \frac{E_{s}}{4N_{0}}} \right)^{{- n} \cdot \delta}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

-   -   where P_(f)(e)_(N) denotes the n-th order approximation equation        of frame error rate (FER),    -   w_(Δ) ^(δ) denotes the number of codewords that have the symbol        Hamming distance δ and the pair-wise product distance Δ from the        all-zero codeword,    -   δ_(min) denotes the minimum symbol Hamming distance,    -   D_(δ) denotes the set of the pair-wise product distances of the        codewords from the all-zero codeword, the codeword having symbol        Hamming distance δ and diverging from the all-zero codeword only        once in trellis diagram,    -   N₀/2 denotes the variance per dimension of AWGN (Additive White        Gaussian Noise),    -   E_(s) denotes the energy per symbol, and    -   n denotes the number of receive antennas.

Hereinafter, the codewords diverging from all-zero codeword only once intrellis diagram are referred to as “codewords simply related to theall-zero codeword”. Therefore, D_(δ) is the set of the pair-wise productdistance of the codeword, which has symbol Hamming distance δ from theall-zero codeword, among the codewords simply related to the all-zerocodeword.

BRIEF DESCRIPTION OF DRAWINGS

The above and other objects, features and advantages of the presentinvention will be more apparent from the following detailed descriptiontaken in conjunction with the accompanying drawings.

FIG. 1A depicts an example of H-symmetric labeling for 4-PSK modulationwith two transmit antennas.

FIG. 1B depicts a conventional H-symmetric labeling.

FIG. 2 is a flow chart depicting a determining process for generatormatrix G in an encoding procedure according to the present invention.

FIG. 3 depicts the frame error rates (FER) performance of geometricalspace-time trellis codes for 4-PSK modulation in a fast fading channelaccording to the present invention.

FIG. 4 depicts the FER performance of geometrical space-time trelliscodes for 4-PSK modulation with 32, 64, and 128 states in the fastfading channel according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, detailed embodiments will be explained for the codingmethod of the present invention with reference to the accompanyingdrawings. However, the invention is not limited by the detailedembodiments below.

In one embodiment, we take into account multiple transceive antennasystem with m transmit antennas and n receive antennas. Further, weassume that we do not know channel state information (CSI) at thetransmitter and we know the channel state information at the receiver.

At time t, it is assumed that there are established m·n channels betweenantenna pairs, wherein m·n channels are independent flat Rayleighfading. The fast fading channels mean that channel gain variesindependently as every symbol.

FIG. 1A shows an example of H-symmetric labeling for 4-PSK modulationwith two transmit antennas according to the present invention.

Labeling means a function mapping an element over alphabet A to bemodulated to a corresponding point in the constellation.

For example, in case of using QPSK (4-PSK) constellation, space-timecodes of prior arts usually use the labeling depicted in FIG. 1B. Thoughthe labeling of FIG. 1B is also an H-symmetry labeling, the labelingcodes of space-time codes in prior arts are not group codes on thealphabet A, which is to be modulated and which is isomorphic with Z₄.

As such, space-time codes of prior arts do not have geometricaluniformity since the labeling codes of space-time codes in the prior artare not group codes on the alphabet A to be modulated.

In the present invention, however, as H-symmetry labeling as depicted inFIG. 1A is used, it is easy to design group codes over alphabet. Thealphabet related to H-symmetry labeling in FIG. 1A is isomorphic to V×V,wherein V is Klein 4-group, and operation × denotes the direct productof groups. In order to perform a systematic code search, we use thegenerator matrix in order to represent group codes over label alphabet.

Assume that a binary sequence of length κ·l denoted by ū=(u₁, . . . ,u_(κ·l)) is transmitted from m antennas, wherein κ=log₂ M, and l is theframe length. Let G be the m×(κ+s) generator matrix for group codes overlabel alphabet, wherein s represents the number of memory elements inthe encoder, each entry being in label alphabet. 2^(s) means the numberof states, and each element of matrix G is one of the elements in thealphabet. Let M be an H-symmetric labeling mapping the coded symbolsover label alphabet to the 4-PSK signal set. We then obtain the codesequence c=(c₁ ¹c₁ ² . . . c₁ ^(m) . . . c_(l) ¹c_(l) ² . . . c_(l)^(m))=( c ₁, . . . , c _(l)) over 4-PSK by applying the H-symmetriclabeling to the following matrix multiplication equation 1:c _(t)=M(ū _(t)

_(Ā) G ^(T))  [Equation 1]

-   -   wherein ū_(t), denotes the input sequence ū_(t)=(u_(κt+(κ−1)) .        . . u_(κt+1)u_(κt) . . . u_(κt−s)) influencing the space-time        symbol c _(t) at time t,    -   _(Ā) is matrix multiplication using addition and multiplication        of Ā,    -   Ā denotes the field whose algebraic structure under addition is        isomorphic to label alphabet A. Then, Ā is isomorphic with        GF(2²)×GF(2²), where GF(2²) is the Galois field of order 2².        Since ū is assumed to be binary sequence, it is natural that the        label code for any generator matrix G over A is a group.

Hereainfter we will explain how to select the generator matrix G.

Assuming that a codeword c was transmitted, the probability that amaximum-likelihood (ML) decoder with perfect CSI prefer another codewordē to c, that is, PEP can be expressed as equation 3.

[Equation 3]

${P\left( \overset{\_}{c}\rightarrow\overset{\_}{e} \right)} \leq {\prod\limits_{t \in {\upsilon{({\overset{\_}{c},\overset{\_}{e}})}}}^{\;}\;{{{{\overset{\_}{c}}_{t} - {\overset{\_}{e}}_{t}}}^{{- 2}n} \cdot \left( \frac{E_{s}}{4N_{0}} \right)^{{- n} \cdot {\delta_{H}{({\overset{\_}{c},\overset{\_}{e}})}}}}}$

-   -   wherein ν( c,ē) denotes the set of time instances tε{1, 2, . . .        , l} such that the Euclidean distance between two space-time        symbols ∥ c _(t)−ē_(t)∥ is nonzero,    -   symbol Hamming distance δ_(H)( c,ē) is the cardinality of ν(        c,ē),    -   E_(s) is the energy per symbol,    -   N₀/2 is the variance per dimension of AWGN, and    -   n is the number of receive antennas.

The previously proposed design criteria for STTCs focused on minimizingthe maximum PEP.

That is, the minimum symbol Hamming distance

$\delta_{H,\min} = {\min\limits_{\overset{\_}{e}}{{\upsilon\left( {\overset{\_}{c},\overset{\_}{e}} \right)}}}$is maximized, and then the minimum product distance

$\Delta_{\min} = {\min\limits_{\overset{\_}{e}}{\prod\limits_{{t \in {\upsilon{({\overset{\_}{c},\overset{\_}{e}})}}},{{{\upsilon{({\overset{\_}{c},\overset{\_}{e}})}}} = \delta_{H,\min}}}^{\;}\;{{{\overset{\_}{c}}_{t} - {\overset{\_}{e}}_{t}}}^{2}}}$is maximized.

However, it is known that the spectrum of the product distancecomparably affects the frame error rate (FER). Thus, we select generatormatrix G considering the distance spectrum.

Let D _(c) be the set of codewords that diverge from the codeword c onlyonce and merge into the codeword c in a trellis diagram. The codewordpair ( c,ē) is called simple if ēεD_(c). Then, FER P_(f)(e) can beexpressed as equation 4.

$\begin{matrix}{{P_{f}(e)} \leq {\sum\limits_{\overset{\_}{c} \in C}^{\;}\;{\sum\limits_{\overset{\_}{e} \in D_{\overset{\_}{c}}}^{\;}{{P\left( \overset{\_}{c}\rightarrow\overset{\_}{e} \right)} \cdot {P\left( \overset{\_}{c} \right)}}}}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

-   -   wherein C denotes the set of codewords,    -   P( c→ē) is PEP, and    -   P( c) is probability that codeword c is transmitted. P( c) may        be 1/|c|, wherein |c| is the number of codewords, since it is        generally assumed that the information occurs with equal        probability.

The conditional FER of geometrically uniform STTC is independent to thetransmitted codeword. Thus, we assume that all-zero codeword 0 wastransmitted. Then, we can obtain following equation 5 from equations 3and 4:

$\begin{matrix}{{P_{f}(e)} \leq {\sum\limits_{\overset{\_}{e} \in D_{\overset{\_}{0}}}^{\;}\;{P\left( \overset{\_}{0}\rightarrow\overset{\_}{e} \right)}} \leq {\sum\limits_{\overset{\_}{e} \in D_{\overset{\_}{0}}}^{\;}{{\Delta\left( {\overset{\_}{0},\overset{\_}{e}} \right)}^{- n} \cdot \left( \frac{E_{s}}{4N_{0}} \right)^{{- n} \cdot {\delta_{H}{({\overset{\_}{0},\overset{\_}{e}})}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

-   -   wherein

${\Delta\left( {\overset{\_}{0},\overset{\_}{e}} \right)} = {\underset{t \in {\upsilon{({\overset{\_}{0},\overset{\_}{e}})}}}{\Pi}{{{\overset{\_}{0}}_{t} - {\overset{\_}{e}}_{t}}}^{2}}$is the pair-wise product distance.

The performance of codes may be estimated from the upper limit of FER ofEquation 5. If the codes are not geometrically uniform, the pair-wiseproduct distance between each of all the codewords and the residualcodewords should be considered. Thus, if the number of codewords is K,the complexity required to estimate the performance of codewordsgenerated from a generator matrix is K(K−1).

However, according to the present invention, codes are geometricallyuniform. Therefore, we have only to calculate the pair-wise productdistances between the all-zero codeword and the residual codewords.Thus, the complexity needed to estimate the performance of codewordsgenerated from a generator matrix is remarkably reduced to K due to thegeometrical uniformity

Let D_(δ) denote the set of the pair-wise product distances of codewordsfrom all-zero codeword 0 wherein the codewords have symbol Hammingdistance δ among the codewords simply related to the all-zero codeword.Let L denote the set of the symbol Hamming distances of codewords fromthe all-zero codeword 0 , wherein the codewords are simply related tothe all-zero codeword. Then equation 5 can be rearranged as equation 2a:

$\begin{matrix}{{P_{f}(e)} \leq {\sum\limits_{\delta \in L}^{\;}\;\left( {\left( {\sum\limits_{\Delta \in D_{\delta}}^{\;}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}} \cdot \frac{E_{s}}{4N_{0}}} \right)^{{- n} \cdot \delta}}} & \left\lbrack {{Equation}\mspace{14mu} 2a} \right\rbrack\end{matrix}$

-   -   wherein w_(Δ) ^(δ) is the number of codewords that have the        symbol Hamming distance δ and the pair-wise product distance Δ        from the all-zero codeword, and that are simply related to the        all-zero codeword. Herein, we refer the term w_(Δ) ^(δ) as        distance spectrum.

Let

$\left( {\sum\limits_{\Delta \in D_{\delta}}^{\;}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}}$be the effective product distance Δ _(δ) _(n) with n receive antennas atthe symbol Hamming distance δ. Examining the equation 2a leads to thefollowing design criteria:

-   -   1) Diversity Criterion: In order to achieve the largest        diversity gain, the minimum Hamming distance should be        maximized; and    -   2) Distance Criterion: In order to achieve the largest coding        gain, the effective product distances at first minimum N symbol        Hamming distances should be serially maximized.

The N denotes the number of the effective product distances. Forexample, at the minimum Hamming distance δ_(min), the effective productdistances with the first N symbol Hamming distances isΔ_(δ) _(min) , Δ_(δ) _(min) +1, . . . , Δ_(δ) _(min) +N−1.

Therefore, the approximate value of FER can be easily calculated basedon the above criteria.

In the present invention, the design criteria that minimize FER will beused. Thus, in order to minimize the value of the equation (i.e., theupper limit of FER), the minimum symbol Hamming distance should bemaximized. Further, after the effective product distances is definedconsidering the distribution of product distances between symbol pairsat random symbol Hamming distances, the generator matrix G thatmaximizes the effective product distances at each symbol Hammingdistances can be easily selected.

Since the maximum PEP used in previous design criteria did not reflectsufficiently FER, the design criteria according to the present inventionto select the generator matrix G are superior.

FIG. 2 is a flow chart that depicts the selecting process of generatormatrix G.

At first, initialize the minimum symbol Hamming distance variable{circumflex over (δ)}_(min) and the effective product distance variableat each symbol Hamming distances {circumflex over (Δ)}_(δ) to be zero(S10).

In order to select the generator matrix G that is suitable to the designcriteria, perform a symbol searching over all the generator matrix G(S20).

Calculate the symbol Hamming distance δ_(min) over the STTCs based on agenerator matrix that are randomly selected (S30).

Compare δ_(min) with the saved value of {circumflex over (δ)}_(min)(S40).

If δ_(min) is smaller than the saved value of {circumflex over(δ)}_(min), perform symbol searching again for another generator matrix(S20).

However, if δ_(min) is larger than the saved value of {circumflex over(δ)}_(min), save the value of δ_(min) into {circumflex over (δ)}_(min)(S50). Therefore, we can select the generator matrix G which maximizesthe value of the symbol Hamming distance δ_(min). At the same time, thecurrent generator matrix is saved, and the {circumflex over (Δ)}_(δ) isinitialized. If the δ_(min) is same with the saved value of {circumflexover (δ)}_(min), it does not matter whether the value of δ_(min) and thecurrent generator matrix is saved or not.

Then, calculate the effective product distance Δ_(δ) at each symbolHamming distance over the STTCs based on the generator matrix G(S60).Compare Δ_(δ) with the value of the saved {circumflex over(Δ)}_(δ)(S70).

If Δ_(δ) is smaller than the saved value of {circumflex over (Δ)}_(δ),select another generator matrix G, and perform symbol searching again(S20).

However, if Δ_(δ) is larger than the saved value of {circumflex over(Δ)}_(δ), save the value of Δ_(δ) at {circumflex over (Δ)}_(δ) (S80). Atthe same time, the current generator matrix is saved. Therefore, we canselect the generator matrix G which maximizes the value of the effectiveproduct distances Δ_(δ) at each symbol Hamming distance.

Then, reiterate symbol searching over other generator matrix G untilsymbol searching over all the generator matrix G is completed (S20).Then, determine the currently stored generator matrix G as the optimalgenerator matrix (S90).

As explained above, in the present invention, we can code messages intogeometrically uniform STTCs, and we can also easily select the optimalgenerator matrix G through the process depicted in FIG. 2. Table I listssome of the search results.

TABLE I 4-PSK GU STTC, 2 bit/s/Hz, frame length 130 symbols. Generatormatrix G δ_(H,min) Δ_(min) ${\overset{\_}{\Delta}}_{\delta_{\min}}^{1}$${\overset{\_}{\Delta}}_{\delta_{\min} + 1}^{1}$ s = 3 $\begin{pmatrix}00 & 10 & 00 & 01 & 11 \\10 & 10 & 01 & 10 & 10\end{pmatrix}\quad$ 2 48 0.3750 0.1163 s = 4 $\begin{pmatrix}00 & 10 & 01 & 10 & 10 & 10 \\10 & 11 & 10 & 00 & 11 & 01\end{pmatrix}\quad$ 3 144 0.3780 0.1780 s = 5 $\begin{pmatrix}00 & 10 & 10 & 01 & 00 & 10 & 01 \\10 & 10 & 11 & 10 & 11 & 10 & 11\end{pmatrix}\quad$ 3 384 3.0236 0.3325 s = 6 $\begin{pmatrix}00 & 10 & 01 & 10 & 10 & 00 & 01 & 11 \\10 & 01 & 10 & 00 & 01 & 10 & 10 & 10\end{pmatrix}\quad$ 4 576 1.9592 0.5800 s = 7† $\begin{pmatrix}00 & 10 & 01 & 01 & 00 & 10 & 00 & 10 & 01 \\10 & 10 & 01 & 10 & 10 & 01 & 01 & 10 & 01\end{pmatrix}\quad$ 4 2304 18.286 1.1180

New GU STTCs generated according to the present invention maximize theminimum Hamming distance and the effective product distances at firstminimum 2 symbol Hamming distances. The symbol † indicates that thesearch for that memory order was not exhaustive.

Firmanto et al. cannot perform the exhaustive code search for 32 states,whereas we can perform the exhaustive code search for 64-states due tothe geometrical uniformity and complexity reduction. Furthermore, it isworthwhile to note that some STTCs with the larger minimum productdistance can be constructed by using a new modulator mapping function,i.e., H-symmetric labeling.

From Firmonto's paper, it is known that if the conventional modulatormapping function of Tarokh is used, the largest achievable minimumproduct distances of the STTCs with 16 and 32 states are 64 and 144.However, according to the present invention, the largest achievableminimum product distances are increased to 144 and 384, respectively.

FIG. 3 shows the FER performance of 4-PSK GU STTC according to thepresent invention with 16 states and 2 bits/s/Hz in fast fadingchannels. TSC, BBH, YB, and FVY STTCs with 16 states are presented forcomparison.

FIG. 4 shows the FER performance of 4-PSK GU STTC according to thepresent invention with 32, 64, and 128 states and 2 bits/s/Hz in fastfading channels. TSC, YB, and FVY STTCs with 32 states are presented forcomparison.

Herein, TSC STTCs means STTCs of Tarokh, et al.; BBH STTCs means STTCsof Bäro, et al. [S. Bäro, G. Bauch, and A. Hansmann, “Improved codes forspace-time trellis-coded modulation,” IEEE Commun. Lett., vol. 4, pp.20-22, January 2000]; YB STTCs means STTCs of Yan, et al. [Q. Yan and R.S. Blum, “Improved space-time convolutional codes for quasi-static slowfading channels,” IEEE Trans. Wireless Commun., vol. 1, pp. 563-571,October 2002]; and FVY STTCs menas STTCs of Firmanto et al.

In both figures, it is observed that codes according to the presentinvention offer a better performance than other known codes.Furthermore, it is worthwhile to mention that 4-PSK 64 and 128-stateSTTCs according to the present invention have a steeper slope than 4-PSK32 states STTCs of the prior art. This occurs because the 64-state and128-state STTCs have δ_(min)=4, while the 32-state STTCs have δ_(min)=3.

As above mentioned, using the design criteria according to the presentinvention, it is possible to provide codes having good performancethrough searching for geometrically uniform space-time codes. Becausethe object of code searching has geometrical uniformity, the complexityof code searching can be reduced remarkably. In addition, due to thisreduction of the complexity, it is possible to search codes which haveSTTCs with more states, and the optimal generator matrix G can be choseneasily. Therefore the STTCs according to the present invention havebetter performance than the STTCs in the prior art.

Although preferred embodiments of the present invention has beendescribed for illustrative purposes, those skilled in the art willappreciate that various modifications, additions and substitutions arepossible, without departing from the scope and spirit of the inventionas disclosed in the accompanying claims.

1. A method for coding a message into space-time trellis codes (STTCs)in a multi-input multi-output (MIMO) communication system, comprisingthe steps of: coding a message with an encoder, into space-time trelliscodes by group codes over label alphabets and H-symmetric labeling,wherein H-symmetric labeling is a modulation function mapping the groupcodes over the label alphabets to a corresponding point in aconstellation; wherein the step of coding includes producing a labelalphabet by matrix-multiplying a binary sequence of the message bygenerator matrix G; and coding the label alphabet into STTCs ^(c) byH-symmetry labeling mapping, wherein the binary sequence u_(t) of themessage at time t is coded into STTC c_(t) by equation 1:c _(t) =M( u_(t) ,

_(Ā) G ^(T))  [Equation 1] wherein ū_(t) is a binary sequence over labelalphabet A that is to be transmitted through m transmit antennas, G is agenerator matrix, where G is dependent on the number of memory elementsin the encoder, Operation {circle around (x)} _(A) is matrixmultiplication defined by addition and multiplication of Ā, Ā denotesthe field whose algebraic structure under addition is isomomhic to thelabel alphabet A, and M is an H-symmetric labeling mapping function. 2.The method of claim 1, wherein there exists a fast fading betweenmultiple transmit antennas and multiple receive antennas.
 3. The methodof claim 2, wherein the coded STTCs have geometrical uniformity.
 4. Themethod of claim 1 further comprising the step of selecting a generatormatrix G which maximizes minimum symbol Hamming distance between acodeword over label alphabet and an all-zero codeword, and alsomaximizes an effective product distance at a predetermined number ofsymbol Hamming distances.
 5. The method of claim 4, wherein thegenerator matrix G is selected to minimize the value of P_(f)(e)_(N) inequation 2: $\begin{matrix}{{P_{f}(e)}_{N} = {\underset{\delta = \delta_{\min}}{\overset{\;}{\sum\limits^{\delta_{\min} + N - 1}}}\;\left( {\left( {\sum\limits_{\Delta \in D_{\delta}}^{\;}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}} \cdot \frac{E_{s}}{4N_{0}}} \right)^{{- n} \cdot \delta}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$ wherein P_(f)(e)_(N) denotes the n-th order approximationof FER (frame error rate), W^(δ) _(Δ) denotes the number of codewordsthat have the symbol Hamming distance δ and the pair-wise productdistance Δ from all-zero codeword, Δ_(min) is the minimum symbol Hammingdistance, D_(δ) denotes the set of the pair-wise product distances ofthe codewords from the all-zero codeword, the codeword having symbolHamming distance δ, and diverging from the all-zero codeword only oncein trellis diagram, N_(0/)2 is the variance per dimension of AWGN, E_(δ)is the energy per symbol, and n is the number of receive antennas.
 6. Amethod for coding a message into space-time trellis codes (STTCs) in amulti-input multi-output communication system, comprising the steps of:selecting among all the generator matrixes, a generator matrix G whichmaximizes minimum symbol Hamming distance between a codeword over labelalphabet and an all-zero codeword, and also maximizes an effectiveproduct distance at a predetermined number of symbol Hamming distances;producing a label alphabet by matrix-multiplying a binary sequence ofthe message by the generator matrix G; and coding the label alphabetwitn an encoder,into STTCs c by H-symmetry labeling which is amodulation function mapping a group code over the label alphabet to acorresponding point in the constellation, wherein the generator matrix Gis dependent on the number of memory element in the encoder,and thegenerator matrix G is selected to minimize the value of P_(f)(e)N inequation 2: $\begin{matrix}{{P_{f}(e)}_{N} = {\sum\limits_{\delta = \delta_{\min}}^{\delta_{\min} + N - 1}\;\left( {\left( {\sum\limits_{\Delta \in D_{\delta}}^{\;}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}} \cdot \frac{E_{s}}{4N_{0}}} \right)^{{- n} \cdot \delta}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$ P _(f(e)) _(N) denotes the n-th order approximation of ERR(frame error rate), W^(δ) _(Δ) denotes the number of codewords that havethe symbol Hamming distance δand the pair-wise product distance Δfromall-zero codeword, δ_(min) is the minimum symbol Hamming distance,D_(δ)denotes the set of the pair-wise product distances of the codewordsfrom the all-zero codeword, the codeword having symbol Hamming distanceδ, and diverging from the all-zero codeword only once in trellisdiagram, N _(0/)2 is the variance per dimension of AWGN, E _(δ)is theenergy per symbol, and n is the number of receive antennas.
 7. A methodfor coding a message into space-time trellis codes (STTCs) in amulti-input multi-output communication system comprising steps of:selecting generator matrix G which minimizes the value of P_(f)(e)_(N)in equation 2; and coding a binary sequence ū_(t) of the message at timet with an encoder into STTC c _(t) by equation 1:c _(t) =M( u _(t)

_(Ā) G ^(T))  [Equation 1] $\begin{matrix}{{P_{f}(e)}_{N} = {\sum\limits_{\delta = \delta_{\min}}^{\delta_{\min} + N - 1}\;\left( {\left( {\sum\limits_{\Delta \in D_{\delta}}^{\;}\;{w_{\Delta}^{\delta} \cdot \Delta^{- n}}} \right)^{{- 1}/{({n \cdot \delta})}} \cdot \frac{E_{s}}{4\; N_{o}}} \right)^{{- n} \cdot \delta}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$ wherein u _(t) is a binary sequence over label alphabet Athat is to be transmitted through m transmit antennas, G is a generatormatrix,where G is dependent on the number of memory elements in theencoder, Operation {circle around (x)} _(A) is matrix multiplicationdefined by addition and multiplication of Ā, Ā denotes the field whosealgebraic structure under addition is isomorphic to the label alphabetA, M is an H-symmetric labeling, which is a modulation function mappingthe group code over label alphabets generated by matrix-multiplying thebinary sequence by generator matrix into signal sequences overconstellation, P_(f(e)) _(N) denotes the n-th order approximation of FER(frame error rate), W_(δ) _(Δ) denotes the number of codewords that havethe symbol Hamming distance δand the pair-wise product distance Δfromall-zero codeword, δ_(min) is the minimum symbol Hamming distance,D_(δ)denotes the set of the pair-wise product distances of the codewordsfrom the all- zero codeword, the codeword having symbol Hamming distanceδ and diverging from the all-zero codeword only once in trellis diagram,N_(0/)2 is the variance per dimension of AWON, E_(δ) the energy persymbol, and n is the number of receive antennas.